nLab Jacobian variety

Idea

To every nonsingular algebraic curve $C$ (over the complex numbers) of genus $g$ one associates the Jacobian variety or simply Jacobian $J(C)$, either via differential 1-forms or equivalently via line bundles: the Jacobian is the moduli space of degree-$0$ line bundles over $C$, i.e. the connected component

$$Jac(X) = Pic_0(X)$$

of the neutral element of the Picard scheme of $C$. See also at intermediate Jacobian – Examples – Jacobian.

Jacobian varieties are the most important class of abelian varieties.

Properties

Abel-Jacobi map

The Abel-Jacobi map $C\to J(C)$ is defined with help of periods.

Line bundles and theta functions

Over the complex numbers, line bundles on a Jacobian variety over a given Riemann surface are naturally encoded by Riemann theta functions.

Related concepts

• Albanese variety

• Prym variety, Prym-Tyurin variety

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

References

• Wikipedia, Jacobian variety, Abel-Jacobi map

• P. Griffiths, J. Harris, Principles of algebraic geometry

• A. Beauville, Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.

A generalizatioin of Abel-Jacobi map to the setting of formal deformation theory is in

• Domenico Fiorenza, Marco Manetti, Formal Abel-Jacobi maps, arxiv/1610.08684

Review for Riemann surfaces includes

• Razvan Gelca, Alejandro Uribe, section 2 of From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)